Artificial selection can only be effective to the extent that an intervention is predictive of the state of the population in the future. The mutual information between artificial selection and the future state of the population quantifies the amount of predictive information [1]} by artificial selection, or alternatively, the memory of the population from the selection intervention.
We characterize the predictive information \(\mathcal {I}_\tau \)  as a time-averaged mutual information \(I(x_{t},x_0)\)  between an intervention (at time \(t=0\) ) and the state of the population at a later time \(t,\, (0<t<\tau )\) , during each intervention cycle in the stationary state,
\(\nonumber \mathcal {I}_\tau &=& \frac{1}{\tau } \int _0^\tau {\text{d}}t\, I (x_{t},x_0) \\\nonumber &= &\frac{1}{\tau } \int {\text{d}}t \int dx_0 dx_t P(x_t, x_0) \log \left[ \frac{P(x_t| x_0)}{P(x_t)}\right]\\\) 
